What You Really Ought to Know

I say life is hard, but optimistically, you say life is simple. I like purple, but you like green. I believe in God, yet you don’t. And you say true love is a prized jewel, but I rebut that true love is just like any other kind of love; only people who were not satisfied with the love they received had adorned it with a more sophisticated and melodramatic look so as to have excuses on the previous precious love that went wrong.   But how could we say which one of us is true?  Why is a truth the truth? This is when guidelines, such as philosophies, shine bright. Mathematics is a big field governed by philosophies and rules that set apart truths from fallacies.

                The book, entitled What is Mathematics, Really? by Reuben Hersh (1997), dealt on the principles that govern mathematics. Platonism, Formalism, Intuitionism, and Hersh’s very own Humanism are the key points discussed in the book.

                One of the most popular philosophies in mathematics was Platonism, which was a work of Plato. Platonism held the belief that mathematical entities originate from an abstract source that is outside man’s reach. He believed that the truths in mathematics are not the products of human minds. Hersh, on the other hand, strongly disagreed with Plato. For Hersh, Platonism simply cannot meet the basic criteria of reality. Empiricism, which is gaining knowledge through experience, is the foundation of science. And with this, Platonism had completely lost the sense of empiricism. I strongly agree with Hersh on this. How would you be able to totally innovate big things, such as mathematical entities, with the absence of first-hand experience or concrete basis? Such as in Chemistry exams, you would never be able to ace or score good in a test just by mentally studying, that is, without any practice.  Even in the other non-mathematical philosophies of Plato, his ideas seem to be…unrealistic.

                Formalism is the next philosophy in line. In Formalism, mathematical truths are arbitrary. That is, the concepts are created without any specific scientific basis or justifiable rules.  Hersh was angered by this because he believed that mathematical entities were created from rules, ideas, and hard works of the men and women that had devoted their lives to Science.  For him, the idea of randomly creating something and without any sense of rational reason is simply unthinkable and worst, non-existent. Hersh, yes, yes, yes. I second the motion. I am a firm believer of the quote “Everything has a reason.”. Well, every invention in this world started only as a response to a stimulus.

                 The next philosophy that Hersh criticized was Intuitionism. The idea of Intuitionism is that Mathematics was developed through mental activity along with the continuous realization of ideas into actual works. Intuitionism emphasizes the importance of intuiting a mathematical entity to give it justification. For with justification comes the basis of existence.

                Among those three major philosophies, namely Platonism, Formalism and Intuitionism, Hersh was convinced by none. Instead, he created his own philosophy, the Humanism. For Hersh, mathematics is a body of knowledge formed in a “social-historic-cultural” way. That is, by the compilation of mathematical entities that had been historically, socially and culturally gained. This consisted of the blood and sweat of our ancestors that had been dedicated for mathematics.  From the word humanism, you could already figure that this pertains to humans: their camaraderie and indescribable efforts resulting to ingenious works.

                In the Humanist view, intuitions lead to ideas that are realized through the interaction of the society. In this interaction, different intellectual ideas and methods are exchanged, and finally, the results of these actions are mathematical concepts that are agreed upon and accepted by the society.

                The concept on infinity is also one of the issues that Hersh acknowledged. Infinities, as we all know, are sets of numbers that do not end. For Hersh, infinity is a fiction. He proclaimed that a finite thing cannot produce infinite objects. His idea is like a cow could never give birth to a pup; or in other words, offsprings are always genetically similar to the parents. Since our minds are finite, therefore, he believed that humans are unable to create something that is infinite. Rather, these “infinite” numbers must only be a concept of infinity, and not really infinity.

                Among all the philosophies discussed, my heart smiles to humanism.  Well, Platonism for me seemed to be unrealistic though its indirect tribute to God is a nice factor. However, everything has its own proper place. Formalism is a no-no since its view on the creation of math by an arbitrary way is like saying “Hey, son. I didn’t mean to create you. You were just an accident.” Oops.

                Intuitionism, on the other hand, is very opposite to Platonism. I agree in Intuitionism on the thought that mathematics is dependent on intellectual capacity and human actions. But what captivated me the most was Hersh’s Humanism.

                Overall, this book has been a great tour guide in my journey into the philosophical world of mathematics. My interest and curiosity in mathematics had been upgraded. With this book,  I have begun to question the things I used to just accept blindly. I have begun to notice the simple things I used to just neglect.  And indeed, math left me craving for more.